Question

There are $3$ candidates for a professorship and one is to be elected by the votes of $5$ men. The number of ways in which the votes can be given is

• A. $125$
• B. $15$
• C. $243$
• D. $8$

Answer 1

Answer: (C)

$243$

There are five ways in which the candidates gets vote from 5 members.

Vote counts

Case 1: 5|0|0

Number of ways $=3$

Case 2: 4|1|0

Number of ways = $(\frac{3}{2})$ $\times 5=3\times 2\times 5=30$ (Assigning candidate $\times$ assigning the voting men)

Case 3: 3|2|0

Number of ways = $(\frac{3}{2})$ $\times \frac{5!}{3!2!}=60$

Case 4: 3|1|1

Number of ways = $(\frac{3}{1})$ $\times \frac{5!}{3!}=60$

Case 5: 2|2|1

Number of ways = $3\times \frac{5!}{2!\times 2!}=90$

Hence the total number of ways = $3+30+60+60+90=243$